Let $ \eta=e^{\frac{2\pi i}n}$ , an $ n$ -th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following product evaluation.

If $ T(n)=\frac{(3n-2)(n-1)}2$ and $ i=\sqrt{-1}$ then $ $ \prod_{j<k}^{0,n-1}(\eta^k-\eta^j)=n^{\frac{n}2}i^{T(n)}.$ $